1、平行線Parallel lines
平行線的性質：兩直線平行，則同位角相等、內錯角相等、同旁內角互補
平行線的判定：同位角相等、內錯角相等、同旁內角互補，則兩直線平行
平行線間的距離：平行線間的距離處處相等（平行線間的平行線段相等）
The properties of parallel lines: if two lines are parallel, the same position angle is equal, the interior angle is equal, and the interior angles on the same side are complementary
Judgment of parallel lines: the same angle is equal, the interior angles are equal, and the interior angles on the same side are complementary, then the two lines are parallel
Distance between parallel lines: The distance between parallel lines is equal everywhere (parallel segments between parallel lines are equal)
關於平行線，需要註意的是，我們常見的幾何圖形中，平行四邊形，長方形，正方形中的平行線的利用。同學們要清楚的知道，在長方形內的三角形，有著大量的同底等高可以利用，在發現同底等高之後，可以輕松地使用面積的等量代換求解問題。
2、三角形(Triangle)
三角形的內角和等於180°（多邊形的內角和：（n-2）×180°）
三角形的外角和等於360°（多邊形的外角和等於360°）
三角形的壹個外角等於與它不相鄰的兩個外角的和 (滿足四點***圓的四邊形，壹個角的外角等於該角的對角的內角。)
三角形的任意兩邊之和大於第三邊，任意兩邊之差等於第三邊 (換算成不等式，可以得到這樣壹個簡單的不等式組，a+b>c, a+c>b,b+c>a, a,b,c,為三角形的三邊長)
The sum of the interior angles of a triangle is equal to 180° (the sum of the interior angles of a polygon: (n-2) × 180°)
The sum of the exterior angles of a triangle is 360° (the sum of the exterior angles of a polygon is 360°)
An exterior angle of a triangle is equal to the sum of the two exterior angles that are not adjacent to it.
The sum of any two sides of a triangle is greater than the third side, and the difference between any two sides is equal to the third side (converted to inequality, such a simple inequality group can be obtained, a+b>c, a+c>b, b+c>a , a,b,c, are the lengths of the three sides of the triangle)
3*、全等三角形
全等三角形的性質：全等三角形的對應角相等，對應邊相等
全等三角形的判定：①S.A.S ②A.S.A ③A.A.S ④S.S.S ⑤H.L
全等三角形的知識點，在IMC的選擇題中不作為重點考察。IMC競賽中，25題均為選擇題，所以我們只需要著重了解全等三角形的性質，註意對應邊全部相等，當發現了兩個三角形全等時，我們不必給出證明，可以直接使用，為後續的題目節約時間。
The knowledge points of congruent triangles are not considered in the multiple-choice questions of IMC. In the IMC competition, all 25 questions are multiple-choice questions, so we only need to focus on understanding the properties of congruent triangles, and pay attention to the fact that the corresponding sides are all equal. Subsequent questions save time.
4、等腰三角形、等邊三角形、直角三角形
等腰三角形兩條腰相等，兩個底角相等
等腰三角形三線合壹：等腰三角形底邊上的高、底邊上的中線、頂角平分線互相重合
等邊三角形的三條邊相等，三個內角等於60°
等邊三角形的判定：有壹個內角等於60°的等腰三角形是等邊三角形
直角三角形的兩個銳角互余
直角三角形兩條直角邊的平方和等於斜邊的平方（勾股定理）
如果三角形兩邊的平方和等於第三邊的平方，那麽這三角形是直角三角形（勾股定理逆定理）
An isosceles triangle has two equal sides and equal base angles
Isosceles triangle three lines in one: the height on the base of the isosceles triangle, the midline on the base, and the bisector of the top angle coincide with each other
The three sides of an equilateral triangle are equal, and the three interior angles are equal to 60°
Judgment of an equilateral triangle: an isosceles triangle with an interior angle equal to 60° is an equilateral triangle
Two acute angles of a right triangle are complementary
The sum of the squares of the two right-angled sides of a right triangle is equal to the square of the hypotenuse (Pythagorean theorem)
If the sum of the squares of the two sides of a triangle is equal to the square of the third side, then the triangle is a right triangle (the inverse of the Pythagorean theorem)
在考試中，看到諸如正六邊形，正方形，壹定要聯想到它們分別與正三角形與等腰直角三角形的聯系(正六邊形可以被簡單的分解為6個小正三角形，正方形可以被看做兩個等腰直角三角形)。並且，同學們需要牢記30°,60°,90°直角三角形的邊長比例關系(正余弦中的特殊值)，以及等腰直角三角形的邊長比例關系(如果已經遺忘，清自行使用勾股定理推導，以便於加深印象)。
總之，幾何問題是IMC競賽考試中很重要的部分，希望同學們能夠結合past paper中的考題，多加復習，爭取在2月2日-3日的考試中獲得高分！